Cohomological Dimension with Respect to Nonabelian Groups

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Doctoral Dissertations Graduate School

8-2006

Cohomological Dimension With Respect to Nonabelian Groups

Atish Jyoti Mitra University of Tennessee - Knoxville

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Recommended Citation Mitra, Atish Jyoti, "Cohomological Dimension With Respect to Nonabelian Groups. " PhD diss., University of Tennessee, 2006. https://trace.tennessee.edu/utk_graddiss/1833

This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a dissertation written by Atish Jyoti Mitra entitled "Cohomological Dimension With Respect to Nonabelian Groups." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics.

Jerzy Dydak, Major Professor

We have read this dissertation and recommend its acceptance:

Robert J. Daverman, Morwen Thistlehwaite, Pavlos Tzermias, John Nolt

Accepted for the Council:

Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by Atish Jyoti Mitra entitled “Co- homological dimension with respect to nonabelian groups.” I have examined the ﬁnal electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy, with a major in mathematics.

Jerzy Dydak Major Professor

We have read this dissertation and recommend its acceptance:

Robert J. Daverman

Morwen Thistlethwaite

Pavlos Tzermias

John Nolt

Accepted for the Council:

Anne Mayhew Vice Chancellor and Dean of Graduate Studies

(Original signatures are on ﬁle with oﬃcial student records) Cohomological dimension with respect to nonabelian groups

A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville

ii Dedication

This dissertation is dedicated to my father Tarak Nath Mitra and to my mother Anjana Mitra, who have always encouraged me to study whatever I liked.

iii Acknowledgments

I want to thank my advisor Prof. Jurek Dydak for everything he has done for me. When I first met him 6 years back I was an outsider to mathematics and he was one of the very few who actually encouraged me to get into graduate studies in mathematics. In the past few years I have got to know him closely, as a teacher, a mentor and a friend. He has redefined for me what teaching and learning mathematics really means. Most important of all, I have always felt that he treated me as a colleague in studying mathematics. Prof. Robert Daverman and Prof. Morwen Thistlethwaite are two other members of the mathematics department whom I was fortunate to have as teachers at the very beginning of my studies. Prof. Daverman’s course in point set topology confirmed my belief in the problem-solving approach to learning mathematics. Prof. Thistlethwaite showed me that algebra can be a very beautiful subject of study by itself and helped me get a lot of geometric flavour out of algebraic topology. Prof. Pavlos Tzermias has been a good friend and great teacher. He helped me learn most of the algebra that I know, and helped me with stimulating expositions on many topics when I was writing my dissertation. I want to thank Prof. John Nolt for agreeing to be a part of my PhD committee. Since Prof. Nikolay Brodskiy joined the mathematics department here he has been a good friend and teacher. I have spent some very enjoyable time studying various topics with him.

iv I spent some nice time hiking in the smokies and discussing mathematics with Prof. Jim Conant. I am grateful to many other teachers in the department who helped me learn mathematics. Prof. Carl Sundberg introduced me to analysis and has been a good friend all these years. Profs. Hinton, Mulay, Rajput, Richter, Simpson, Wade have been good mentors. Fellow students have made the mathematics department a good place to be in. I want to mention Erika Asano, Wandi Ding, Volodymyr Hrynkiv, Violeta Vasilevska and Mike Saum - all of whom I have known for most of my stay here. My brother Niloy has been a constant source of encouragement during the past few years. Any eﬀort to express my gratitude towards my parents will be insuﬃcient. They have been very supportive of me all these years while I was enjoying the luxury of graduate studies.

v Abstract

This dissertation addresses three aspects of cohomological dimension of metric spaces with respect to nonabelian groups. In the first part we examine when the Eilenberg-Maclane space (n = 1) of the abelianization of a solvable group being an absolute extensor of a metric space implies the Eilenberg-Maclane space of the group itself is an absolute extensor. We also give an elementary approach to this problem in the case of nilpotent groups and 2-dimensional metric spaces. The next part of the dissertation is devoted to generalizations of the Cencelj- Dranishnikov theorems relating extension properties of nilpotent CW complexes to its homology groups. In the final part we extend the definition of Bockstein basis of abelian groups to nilpotent groups G, and prove a version of the First Bockstein Theorem for such groups.

vi Contents

1 Introduction 1

2 Cohomological dimension theory 3 2.1 Covering dimension ...... 3 2.2 Cohomological dimension ...... 4 2.3 Extension theory ...... 5

3 Nilpotent and solvable groups 7 3.1 Abelian series and solvable groups ...... 7 3.2 Central series and nilpotent groups ...... 8 3.3 Abelianizations of nilpotent groups ...... 9

4 Dimension with respect to groups and their abelianizations 11 4.1 Dimension with respect to nonabelian groups ...... 11 4.2 Preliminaries ...... 14 4.3 Nilpotent groups ...... 15 4.4 Finite solvable groups ...... 16 4.5 Extension properties of the Pontryagin Disk ...... 19 4.6 Inﬁnite solvable groups ...... 20

5 Hurewicz-Serre theorem for nilpotent groups 23 5.1 Introduction ...... 23

vii 5.2 Properties of the homotopy ﬁber of L → SP (L) ...... 25 5.3 Homotopy groups with coeﬃcients ...... 27 5.4 Main results ...... 31 5.5 Appendix on nilpotent groups ...... 34

6 Bockstein Theorem for nilpotent groups 37 6.1 Introduction ...... 37 6.2 Exact sequences ...... 39 6.3 Nilpotent groups ...... 40 6.4 Bockstein basis ...... 42 6.5 Bockstein spaces ...... 47

Bibliography 50

Vita 56

viii Chapter 1

Introduction

One of the most basic geometric concepts is that of dimension. It is a concept that is as intuitive as it is difficult to define in a rigorous way. One first encounters the notion of dimension in a purely algebraic setting in elementary linear algebra and gets the algebraic dimension of the line and plane. The first non-algebraic ways of thinking about dimension comes from elementary point set topology. In general topology there are various ways of defining dimension (small inductive dimension, large inductive dimension, covering dimension, etc.), most of which coincide in nice spaces. There are nice expositions of these notions in [32]. Once homology and cohomology groups are defined it is useful to introduce some algebra into the geometric setting of dimension. Then we have the notions of homological and cohomological dimension with respect to Z, and soon can generalize to homological and cohomological dimension with respect to arbitrary abelian groups. These questions have kept topologists busy for many years, maybe starting from Alexandroff’s questions [1] during the First International Topology Conference in Moscow in 1935. Kuzminov’s influential paper [41] of 1968 was a great contribution to the algebraic point of view in studying homological dimension. A return to the

1 geometric roots of cohomological dimension followed in 1981 starting with Walsh’s el- egant paper [49]. Since then Dranishnikov solved many of the problems of homological and cohomological dimension theory [18, 19, 20]. Dydak has major contributions in cohomological dimension and extended many of the results from the case of compacta to arbitrary metric spaces [25, 28]. The first attempt to study cohomological dimension of compacta with respect to non-abelian groups was by Dranishnikov and Repovˇsin [23] - they considered the case of perfect groups and used geometric techniques. Some of the ideas were explored later in [11]. Later Cencelj and Dranishnikov looked at the case of nilpotent groups to answer some questions - specifically attempting to develop a Hurewicz-Serre type theorem for nilpotent complexes [5, 6, 7]. In this dissertation we look at some questions concerning cohomological dimension of metrizable spaces with respect to nilpotent and solvable groups. After some back- ground on cohomological dimension and groups in the next two chapters, in chapter 4 we consider the problem of comparing cohomological dimension of spaces with respect to groups and their abelianizations. We provide generalizations to some existing results [5, 6, 7] of nilpotent groups and then move to the case of solvable groups. In chapter 5 we provide a version of Hurewicz-Serre theorem for nilpotent groups. In chapter 6 we prove a version of the Bockstein first theorem for nilpotent groups.

2 Chapter 2

Cohomological dimension theory

2.1 Covering dimension

Deﬁnition 2.1.1. Let X be a normal space. The covering dimension dimX (also known as the Cech-Lebesgueˇ dimension) is deﬁned as follows:

1. dimX ≤ n, where 0 ≤ n < ∞, if every finite open cover of X has a finite open refinement of order n (every n + 2 distinct elements of the cover have empty intersection)

2. dimX = n,where 0 ≤ n < ∞, if dimX ≤ n and dimX n − 1

3. dimX = ∞, if dimX n for all n.

There are various other ways of deﬁning dimension (the large inductive dimension and small inductive dimension) and we refer the reader to [32] for a discussion of cases in which the various notions coincide. For example, they all coincide for seper- able metrizable spaces. In Euclidean spaces the topological and algebraic dimensions coincide.

Theorem 2.1.2. The algebraic and topological notions of dimension coincide for Rn.

3 2.2 Cohomological dimension

The following theorem of Alexandroﬀ lets us think of covering dimension in an algebraic way.

Theorem 2.2.1. If X is a ﬁnite dimensional compactum, then its dimension is the

smallest integer n such that Hˇ n+1(X,A; Z) = 0 for all closed subsets A of X.

In the above theorem one can replace Z by any abelian group G 6= 0 which leads

to the notion of the cohomological dimension dimG(X).

Deﬁnition 2.2.2. Given a paracompact space X and an abelian group G, one assigns to X the cohomological dimension dimG(X) which is an integer larger than or equal to -1, or inﬁnity (∞), by the following conditions:

ˇ n+1 1. dimG(X) ≤ n, where 0 ≤ n < ∞, if H (X,A; G) = 0 for all closed subsets A of X,

3. dimG(X) = n, where 0 ≤ n < ∞, if dimG(X) ≤ n and dimG(X) ≤ n − 1 does not hold,

4. dimG(X) = ∞, if dimG(X) ≤ n does not hold for all n < ∞.

The relevance of dimension with respect to arbitrary groups was underscored by Pontryagin [46] who realized that in order to construct compacta X and Y with dim(X × Y ) < dim(X) + dim(Y ), one has to construct a compactum P such that dim(P ) > dimZ/p(P ) for some prime p.

Theorem 2.2.3. For each prime p there is a compactum Pp such that the following conditions hold:

1. dim(Pp) = 2, dimZ/q(Pp) = 1 if q is a prime and q 6= p,

2. dim(Pp × Pq) = 3 if q is a prime and p 6= q.

4 2.3 Extension theory

Deﬁnition 2.3.1. K is an absolute extensor of X (notations: K ∈ AE(X) or XτK) if every map f : A → K extends over X if A is closed in X.

We can use absolute extensors to characterize covering dimension.

Theorem 2.3.2. If X is compact, then dim(X) ≤ n if and only if Sn ∈ AE(X).

A similar characterization holds in cohomological dimension.

Theorem 2.3.3. [Cohen’s Theorem] If X is locally compact, then dimG(X) ≤ n if and only if K(G, n) ∈ AE(X).

When translating results from extension theory to cohomological dimension theory we ﬁnd that the following theorem is of fundamental importance.

Theorem 2.3.4. Suppose X is a metrizable space and K is a connected CW complex. Consider the following conditions:

1. K ∈ AE(X).

2. SP ∞(K) ∈ AE(X).

3. dimHm(K;Z)(X) ≤ m for all m ≥ 0.

4. dimπm(K;Z)(X) ≤ m for all m ≥ 0.

Then, 1 implies 2. If K is simply connected, then Conditions 2-3-4 are equivalent. If X is of ﬁnite dimension and K is simply connected, then Conditions 1-2-3-4 are equivalent.

Recall that SP ∞(K) is the inﬁnite symmetric product of K (see [16]). For X compact, Theorem 2.3.4 is due to A.Dranishnikov [19]. Subsequently, it was generalized to metrizable spaces by J.Dydak [25], [28].

5 Theorem 2.3.4 is reminiscent of the classical Hurewicz Theorem, and in fact part of its proof relies on the Serre version of the Hurewicz Theorem. Thus, it represents a point of overlap between extension theory and algebraic topology. A geometric result repeatedly used in the next chapters is the following.

Theorem 2.3.5. Let X be a metric space and let L, K be complexes. If F is the homotopy ﬁber of L → K and XτF , then XτL is equivalent to XτK.

6 Chapter 3

Nilpotent and solvable groups

A natural way to build groups out of abelian groups is by forming extensions. Nilpo- tent and solvable groups arise in this way.

3.1 Abelian series and solvable groups

Deﬁnition 3.1.1. 1. A normal series of a group is a ﬁnite sequence of subgroups

1 = G0 ¡ G1 ¡ ··· ¡ Gn = G such that Gi ¡ G for all i.

2. An abelian series is a normal series in which each factor Gi+1/Gi is abelian.

Deﬁnition 3.1.2. A group is called solvable if it has an abelian series.

Every abelian group is trivially solvable. The simplest example of a nonabelian solvable group is the symmetric group S3.

Deﬁnition 3.1.3. 1. The length of a normal series is the number of nontrivial factor groups.

2. For any solvable group G, the length of the shortest abelian series is called the derived length.

7 Thus only the trivial group has derived length 0, and abelian groups are the groups of derived length 1. We will now describe a particularly nice way of constructing an abelian series that realizes the derived length of a solvable group. For any group G we can construct the derived subgroup G0 = [G, G], the subgroup generated by all commutators in G. By repeatedly forming derived subgroups we get a descending sequence G = G(0) > G(1) > ··· , where G(n+1) = (G(n))0 . Clearly all the factors are abelian, but in general the series may not terminate, unless the group is solvable as in the next result.

Proposition 3.1.4. If 1 = G0 ¡ G1 ¡ ··· ¡ Gn = G is an abelian series of a solvable

(i) group, then G < Gn−i, and the derived length of the group is equal to the length of the derived series.

The next fact shows that solvable groups form a nice class.

Proposition 3.1.5. The class of solvable groups is closed with respect to formation of subgroups, images and extensions.

Finite solvable groups have particularly nice abelian series.

Proposition 3.1.6. A ﬁnite group is solvable if and only if it has a series with all factors cyclic of prime order.

3.2 Central series and nilpotent groups

Deﬁnition 3.2.1. A central series is an abelian series in which each Gi+1/Gi is

contained in the center of G/Gi.

Deﬁnition 3.2.2. A group is called nilpotent if it has a central series.

As in the case of solvable groups, the next result shows that nilpotent groups form a nice class.

8 Proposition 3.2.3. The class of nilpotent groups is closed with respect to formation of subgroups, images and ﬁnite direct products.

Finite nilpotent groups can be characterized as below.

Proposition 3.2.4. A ﬁnite group is nilpotent if and only if it is the direct product of its Sylow subgroups.

3.3 Abelianizations of nilpotent groups

This section develops a nice technique that lets us conclude facts about nilpotent groups by studying their abelianizations.

Recall that the lower central series of a group G is G = γ1G = γ2G = ··· , where

γi+1G = [γiG, G]. Notice that γiG/γi+1G lies in the center of G/γi+1G and that the nilpotent class of G equals the length of the lower central series of G. The following lemma shows that the ﬁrst lower central factor Gab exerts a strong inﬂuence on the subsequent lower central factors.

Lemma 3.3.1 (Robinson). Let G be a group and let Fi = γiG/γi+1G. Then the mapping a(γi+1G) ⊗ g[G, G] 7→ [a, g](γi+2G) is a well deﬁned module epimorphism from Fi ⊗Z Gab to Fi+1.

Proof: Let g ∈ G and a ∈ Gi, and consider the function (a(γi+1G), g[G, G]) 7→

[a, g](γi+2G). Check that the mapping is well-deﬁned and bilinear. By the universal property of tensor products, we have an induced homomorphism. It is easily checked that it is an epimorphism.

The above lemma gives the following useful corollary.

Corollary 3.3.2. Let P be a group theoretical property which is inherited by images of tensor products (of abelian groups) and by extensions. If G is a nilpotent group such that Gab has P, then G has P.

9 Proof: Let Fi = γiG/γi+1G, and note that F1 = Gab. Suppose Fi has P : then

the previous lemma implies that Fi+1 has P. As G is nilpotent, some γc+1 = 1. Since P is closed under extensions, G has P.

Corollary 3.3.3. 1. If G is a nilpotent group and Gab is ﬁnite , then so is G.

2. If G is a nilpotent group and Gab is p-torsion , then so is G.

10 Chapter 4

Dimension with respect to groups and their abelianizations

In chapter 2 we gave a brief outline of cohomological dimension of metric spaces. In this chapter we will concentrate on the more geometric viewpoint of looking at cohomological dimension in terms of extensions of maps. Recall that for any compactum

X and abelian group G, dimGX ≤ n ⇔ XτK(G, n), where τ is Kuratowski’s notation described in chapter 2 and K(G, n) is the Eilenberg-Maclane space.

4.1 Dimension with respect to nonabelian groups

Trying to extend the notion of cohomological dimension to non-abelian groups G makes sense only for n = 1, as the corresponding Eilenberg-Maclane spaces are deﬁned only for n = 1. Dranishnikov and Rˇepovs studied cohomological dimension with respect to perfect groups in [23], using techniques of a geometric ﬂavour. Later Cencelj and Dranishnikov studied the case of nilpotent groups in a series of three papers [5, 6, 7].

11 Generalizing Dranishnikov’s theorem 2.3.4 about extension of maps to simply- connected complexes [19], they obtained the following result.

Theorem 4.1.1. For any nilpotent CW-complex M and ﬁnite-dimensional metric compactum X, the following are equivalent:

1. XτM

2. XτSP∞M

3. dimHi(M)X ≤ i for every i > 0

4. dimπi(M)X ≤ i for every i > 0

i i Recall that SP X = X /(Σi), where Σi is the symmetric group on i letters. If X is pointed, there is a natural embedding SPiX → SPi+1X, and SP∞X is deﬁned as direct limit of the SPiX. In course of the proof of 4.1.1, they obtained the following relation between dimension with respect to a nilpotent group and its abelianization.

Theorem 4.1.2. For a nilpotent group N and every metric compactum X the fol-

lowing equivalence holds : dimN X ≤ 1 ⇔ dimNab X ≤ 1 provided N has one of the following properties:

1. N is a torsion group.

2. for every prime p such that TorpN 6= 1

(a) N is not p-divisible, or

(b) TorpNab 6= 0.

In view of 4.1.2 it seems natural to ask the following question.

Question 4.1.3. Let C be a class of spaces. Describe groups for which the following are equivalent for any X ∈ C:

12 1. XτK(G, 1)

2. XτK(Gab, 1)

The following theorem gives one of the implications of question 4.1.3 for any group G.

Theorem 4.1.4. Let X be a metric space and G a group. Then XτK(G, 1) ⇒

XτK(Gab, 1).

Theorem 4.1.4 was proven for the case of all compacta by Dranishnikov and Cencelj in [5]. The proof of theorem 4.1.4 in its present form is essentially contained in a paper by Dydak [29]. The proof depends on two results, the ﬁrst of which is due to Dydak and the second is a version of the celebrated Dold-Thom theorem.

Lemma 4.1.5. Suppose X is metrizable, K is a pointed CW complex, and XτK. Given a closed subset A of X and g : A → SPk(K), there is an extension g0 : X → SPk.k!(K).

Theorem 4.1.6. [16] Let K be a CW complex. Then SP∞K has the weak homotopy Q type of a product of Eilenberg-Maclane spaces K(Hn(K), n).

Proof. (of theorem 4.1.4) Replacing SP∞(K(G, 1)) by the telescope S SPk(K(G, 1))× [k − 1, k] and using 4.1.5 we have XτSP∞K(G, 1). Then use 4.1.6 to see that

XτK(H1(K(G, 1)), 1), which is equivalent to XτK(Gab, 1). In the next section ( 4.2) we list well known results which will be useful later in the chapter. In section 4.3 we show that if C is the class of all 2-dimensional compacta, then nilpotent groups satisfy question 4.1.3. In section 4.4 we show that if C is the class of all 2-dimensional compacta, then ﬁnite solvable groups satisfy question 4.1.3. Finally in section 4.6 we give an example showing that inﬁnite solvable groups do not satisfy question 4.1.3 even when C is the class of all 2-dimensional compacta.

13 4.2 Preliminaries

We collect below some classical results which we will use in the rest of the chapter.

Proposition 4.2.1. Let X and Y be metric spaces and let A be closed in X. A map f : A → Y has an extension to X iﬀ some map g : A → Y homotopic to f has an extension.

Recall that a map f : X → Y between CW complexes is called cellular if f(X(n)) ⊂ Y (n) for all n.

Theorem 4.2.2 (Cellular Approximation Theorem). [42] Every map f : X → Y between CW complexes is homotopic to a cellular map.

Theorem 4.2.3 (Seifert - Van Kampen ). Let X1, X2 be path connected open

subsets of X, and the intersection X0 = X1 ∩X2 be path connected. Then the following commutative diagram

i1 π1(X0) −−−→ π1(X1)    j i2y y 1

j2 π1(X2) −−−→ π1(X)

of fundamental groups (based at some x0 ∈ X0) where all maps are induced by inclusions is a pushout diagram.

Theorem 4.2.4 (Bockstein exact sequence). [25] If 1 → G → E → Π → 1 is a short exact sequence of abelian groups, then there is a natural exact sequence

· · · → Hˇ n(X,A; G) → Hˇ n(X,A; E) → Hˇ n(X,A; Π) → Hˇ n+1(X,A; G) → · · · (4.1)

for any paracompact space X and its closed subspace A. Here Hˇ n(X,A; G) is the n-th Cechˇ cohomology group of the pair (X,A) with coeﬃcients in G.

14 4.3 Nilpotent groups

We present a minor generalization of theorem 4.1.2 for 2-dimensional metrizable spaces, with an elementary proof.

Theorem 4.3.1. Let G be a nilpotent group and X be a 2-dimensional metric space.

Then XτK(G, 1) ⇔ XτK(Gab, 1).

Recall from chapter 3 the useful device 3.3.2 of proving statements about nilpotent groups from their abelianizations: if P be a group theoretical property which is inherited by images of tensor products (of abelian groups) and by extensions, and if

G is a nilpotent group such that Gab has P, then G has P. Proof. ( of theorem 4.3.1) Let P be the group theoretical property of ”belonging to G”, the class of groups with XτK(G, 1). We know that if 1 → Γ → E → Π → 1 is a short exact sequence and XτK(Γ, 1) and XτK(Π, 1), then XτK(E, 1)[25]. We also know that if X is metrizable, then XτK(Γ, 1) ⇒ XτK(Γ ⊗ Π, 1) for any abelian groups Γ, Π [25]. Finally we consider the Bockstein exact sequence 4.2.4 for the short exact sequence 1 → ker φ → Γ → φ(Γ) for any homomorphism φ and any abelian group Γ . As X is 2-dimensional, Hˇ 3(X,A; ker φ) = 0. Now if XτK(Γ, 1) then Hˇ 2(X,A; Γ) = 0, so the Bockstein sequence reduces to 0 → Hˇ 2(X,A; φ(Γ)) → 0, which shows Xτ(K(φ(Γ), 1)). Then we apply 3.3.2 to complete the proof.

Remark 4.3.2. In the above that proof we only need X to be a metric space of cohomological dimension 2.

15 4.4 Finite solvable groups

Our eﬀort to answer question 4.1.3 for solvable groups gives the next theorem. The nice trick 3.3.2 that we used for nilpotent groups does not work for solvable groups, hence we have to use geometric methods for proving 4.4.1.

Theorem 4.4.1. Let G be a ﬁnite solvable group and X be a 2-dimensional metrizable

space. Then XτK(G, 1) ⇔ XτK(Gab, 1).

For proving the above theorem, we need some preliminaries. The following result from extension theory must have been well known. As we couldn’t locate a proof in the usual texts, we outline a proof.

Proposition 4.4.2. Let X be a metrizable space and K1, K2 be subcomplexes of the

CW complex K1 ∪ K2. Then:

1. If K1, K2 and K0 = K1 ∩ K2 are absolute extensors of X, then so is K1 ∪ K2.

2. If K1 ∪ K2 and K0 = K1 ∩ K2 are absolute extensors of X, then so are K1, K2.

Proof. (1) Let f : A → K1 ∪ K2 be a map from a closed subset A of X. Deﬁne

−1 Ci = f (Ki) and note that A = C1 ∪ C2 and f(C1 ∩ C2) ⊂ K1 ∩ K2. As the closure

of any one of the sets C1 − C2 and C2 − C1 misses the other we can ﬁnd U open in

X such that C1 − C2 ⊂ U ⊂ U ⊂ X − (C2 − C1). Deﬁne D1 = U ∪ (C1 ∩ C2) and

D2 = (X −U)∪(C1 ∩C2). Then we have D1 ∩A = C1, D2 ∩A = C2 and D1 ∪D2 = X.

As C1 ∩ C2 is closed in D1 ∩ D2 (which is closed in X) and as K1 ∩ K2 is absolute extensor of X, we can extend f | C1 ∩ C2 to f : D1 ∩ D2 → K1 ∩ K2. Now we have a map from C1 ∪ (D1 ∩ D2) to K1 and a map from C2 ∪ (D1 ∩ D2) to K2, both of which we can extend and then paste the resulting maps to get the desired extension.

(2) Start with f : A → K1 and extend to f : X → K1 ∪ K2. Calling Ci =

−1 (f) (Ki), we have X = C1 ∪ C2, A ⊂ C1 and f(C1 ∩ C2) ⊂ K1 ∩ K2. Extend

f | C1 ∩ C2 to fe : C2 → K1 ∩ K2. Paste f | C1 and fe | C2 to get the desired extension.

Deﬁnition 4.4.3. Let G1, G2 and A be groups, and fi : A → Gi be homomorphisms. G1 ∗ G2 Then the amalgamated product G1 ∗A G2 is deﬁned as (Here hf1(a) = f2(a), ∀a ∈ Ai hi denotes the normal closure). Notice that the amalgamated product is a pushout of the following diagram:

f1 A −−−→ G1  f  2y

The following proposition is of interest in itself. We use it later to get a useful lemma.

Proposition 4.4.4. Let X be a 2-dimensional metrizable space and deﬁne HX = {G : XτK(G, 1)}.

1. If G1,G2 and A are in HX , then so is G1 ∗A G2.

2. If G1 ∗A G2 and A are in HX , then so are G1 and G2.

Proof. Note that a homomorphism fi : A → Gi, i = 1, 2 induces a map f∗i :

(2) (2) K(A, 1) → K(Gi, 1) : we start by sending the generators to the corresponding

(1) loops in K(Gi, 1) , and then use the relators of A to extend the map to the 2-skeleton.

(2) Let Mf∗i be the mapping cylinder of f∗i: then we have K(A, 1) ,→ Mf∗i. Finally

(2) we apply the previous theorem with K1 = Mf∗1, K2 = Mf∗2 and K0 = K(A, 1) .

We can now prove the following lemma, which we use later to prove theorem 4.4.1.

17 Lemma 4.4.5. Let X be a 2-dimensional metrizable space with dim Z/pX = 1 and p let K be a CW complex. Let α ∈ π1(K) with α = 1, for some prime p. Then

2 2 XτK ⇔ XτK ∪α D , where D is a 2-cell.

Proof.

1 p Note that α : S → K induces a homomorphism α∗ : Z → π1(K) with α∗(1) = 1,

2 i.e. α∗(p) = 1. Then π1(K ∪α D ) is the pushout of

α1 Z −−−→ π1(K)   y {1} But in this case it is also the pushout of

α1| Z/p −−−→ π1(K)   y {1} Then we can use proposition 4.4.4.

The previous lemma has the following generalization:

Lemma 4.4.6. Let X be a 2-dimensional metrizable space, P = {p : dim Z/pX = 1}

m pi i and let K be a CW complex. Let αi ∈ π1(K) with αi = 1, for some prime pi ∈ P

2 2 and positive integer mi. Then XτK ⇔ XτK ∪αi Di , where Di is a 2-cell.

Proof. The proof uses induction on the number of primes and the same technique used in 4.4.5.

Proof. (of theorem 4.4.1)

18 n1 nl A = Gab is a ﬁnite abelian group and hence of the form Z/q1 ⊕ · · · ⊕ Z/ql .

Using the previous lemma, we can ”kill” all elements of G of order products of qi: we create the quotient of G by the normal subgroups generated by those elements. As G is ﬁnite , we can repeat this process ﬁnitely many times to get a solvable group G˜ such that G˜ has no elements of order products of primes appearing in the decomposition ˜ ˜ of Gab into cyclic groups, and XτK(G, 1) ⇔ XτK(G, 1) for 2-dimensional compacta X. We claim that G˜ has trivial abelianization: in the exact sequence 1 → [G,˜ G˜] →

˜ ˜ ˜ 0 q G → Gab → 1 let y be any generator of Gab and consider its preimage y . As y = 1 0 q ˜ 0 q for some q ∈ {q1, ··· , ql}, y is mapped to 1 in Gab, and hence y = k for some ˜ ˜ ˜ ˜ element k ∈ [G, G]. Every element in [G, G] has order relatively prime to the qi(s),

0 pq so suppose k has order p, relatively prime to the qi(s). Thus we have y = 1 which

0 p implies y = 1, which in turn implies yp = 1. Thus y = 1, and our claim is proved.

4.5 Extension properties of the Pontryagin Disk

We will use in this section a 2-dimensional compactum called the mod-2 Pontryagin

2 disk PD , which is a variant of a classical construction of L.S. Pontryagin [46]. 2 The space PD is constructed as the inverse limit of an inverse sequence of spaces. As the ﬁrst stage of the construction, we start with the usual 2-cell embedded in R3 and take some triangulation of it with mesh less than 1. To get the k+1-th stage from the k-th stage we take each 2-simplex, remove its interior and glue a disk along the degree 2 map, then give a triangulation of mesh less than 1/(k + 1) to the resulting space. For the bonding map from k+1-th stage to the k-th stage, we send the interior 2-simplices of the attached disks to the barycenter of the removed 2-simplices.

19 The following theorem is very useful in constructing compacta with special exten-

sion properties. For explanation of terms and proof see [23]. Note that mesh{λi} → 0 means that for every k,

k+i limi→∞{mesh(qk (λ)k+i)} = 0. This is not the same as limi→∞meshλi = 0.

Theorem 4.5.1. Suppose that K is a countable CW complex and that X is a com-

i+1 i+1 pactum such that X = lim←(Xi, fi ), where (Li, fi ) is a K-resolvable inverse system of compact polyhedra Li with triangulations λi such that mesh{λi} → 0. Then XτK.

2 2 Theorem 4.5.2. Let PD be the Pontryagin disk. Then PD τK(Z/2, 1).

2 Proof. Let A ⊂ PD be a closed and consider the map f : A → K(Z/2, 1). 2 2 i+1 Represent (PD ,A) = lim←{((PD )i,Ai), qi } in the usual way as the inverse limit

∞ of polyhedral pairs (all lying in I ). Find i0 such that f can be extended over Ai0 to

∞ give a map fi0 : Ai0 → K(Z/2, 1) with fi0 ◦(qi0 ) ' f. Extend fi0 over the 1-skeleton of 2 2 (1) ((PD )i and get an induced extension up to homotopy G : A ∪ (PD ) → K(Z/2, 1).

2 (1) (1) Pick any 2-simplex σj ∈ ((PD )i0 ) − (Ai0 ) and consider γσj = gi0 (∂σj) ∈ Z/2.

∞ −1 2 Consider (qi0 ) (σj) ⊂ PD and note that we have an extension (of f) over A ∪ ∞ −1 ∞ −1 (qi0 ) (∂σj). We want to extend over A ∪ (qi0 ) (σj): if γσj is trivial there is no obstruction. If not, ﬁnd the smallest index i > i such that (pi1 )−1 has an ( P )2. 1 0 i0 R ∞ −1 Clearly the extension over A∪(qi0 ) is possible. Do the same for all such 2-simplices

σj.

4.6 Inﬁnite solvable groups

2 Theorem 4.6.1. Let PD be the Pontryagin disk and Γ be a torsion free group with

abelianization being a non-trivial 2-group. Then K(Γab, 1) is an absolute extensor of

2 2 PD , but K(Γ, 1) is not an absolute extensor of PD .

20 2 Proof. Let σ be any 2-simplex in the ﬁrst stage of construction of PD and

∞ −1 identify ∂σ and its image in X under (f0 ) ∂σ. Consider the map f : ∂σ → K(Γ, 1) that sends ∂σ to the generator x along the identity map S1 → S1.

2 We claim that this map does not extend: suppose there is an extension g : PD →

K(Γ, 1). Then there is some integer i0 such that there is an extension up to homotopy

2 gi0 : PD i0 → K(Γ, 1). In that case, x could be expressed as products of elements of Γ, each of order 2. But Γ is torsion-free, and we have a contradiction.

We give 2 examples of solvable groups satisfying 4.6.1, with Γab being a 2-group.

Example 4.6.2. Consider the following group, which was ﬁrst studied by Hirsch:

z −1 z −1 4 Γ1 = hx, y, z|x = x , y = y , [x, y] = z i. This example of Hirsch is a torsion free polycyclic group which is not poly inﬁnite cyclic. The folowing lemma shows that G is not nilpotent.

Lemma 4.6.3 (Robinson, 5.2.20). A ﬁnitely generated torsion-free nilpotent group has a central series with inﬁnite cyclic factors.

Example 4.6.4. 1 It is known [51] theorem 3.5.5 that there are just 6 aﬃne dif- feomorphism classes of compact connected orientable ﬂat 3-dimensional Riemannian

manifolds, represented by manifolds R3/G, where G is one of 6 known groups. One of them is our next example

2 −1 −1 −1 −1 Γ2 = hα, β, γx, y, z|γβα = xy, α = x, αyα = y , αzα = z , βxβ−1 = x−1, β2 = y, βzβ−1 = z−1, γxγ−1 = x−1, γyγ−1 = y−1, γ2 = zi.

Its abelianization is Z/4 ⊕ Z/4. It is torsion free and solvable but not nilpotent. 1The author thanks Prof. Thomas Farrell for pointing out this example.

21 Remark 4.6.5. Even solvable groups with some elements of inﬁnite order may not be enough to serve as an example of groups in 4.6.1.

As an example, consider D∞, which is deﬁned to be the semi-direct product Z oθ

−1 Z/2, where θ : Z/2 → Aut(Z) is the ”t ” map. A presentation of D∞ is hx, y |

2 −1 −1 2 y = 1, yxy = x i. Note that the relations imply (yx) = 1, so D∞ = Z/2 ∗

Z/2 and therefore, the abelianization is Z/2 ⊕ Z/2. As D∞ = Z/2 ∗ Z/2, we have 2 PD τK(D∞, 1).

22 Chapter 5

Hurewicz-Serre theorem for nilpotent groups

5.1 Introduction

Given a metrizable space X and a connected CW complex L consider the following conditions:

1. XτL.

2. XτK(πn(L), n) for all n ≥ 1.

3. XτK(Hn(L), n) for all n ≥ 1.

In chapter 2 we pointed out that (1) implies (3) was proved by Dranishnikov [19] for X compact and for arbitrary X in [[25], Theorem 3.4]. The diﬃculty in generalizing results in cohomological dimension theory from compact spaces to arbitrary metrizable spaces usually lies in the fact that the First Bockstein Theorem does not hold for metric spaces.

0This chapter is based on joint work with Cencelj,Dydak and Vavpetiˇc[8]

23 By a Hurewicz-Serre Theorem in Extension Theory we mean any result showing (3) implies (2). Here is the main problem for this chapter:

Problem 5.1.1. Suppose X is a metrizable space such that XτK(Hn(L), n) for all n ≥ 1. If L is nilpotent, does XτK(πn(L), n) hold for all n ≥ 1?

Even a specialized version of 5.1.1 is open:

Problem 5.1.2. Suppose L is a nilpotent CW complex. If X is a metrizable space

such that XτK(Hk(L), k) for all k ≥ 1, does XτK(π1(L), 1) hold?

Notice that it is not suﬃcient to assume XτK(H1(L), 1) in 5.1.2. Namely, take

the group G from [50] whose abelianization is Q ⊕ Q and whose commutator group

∞ ∞ is Z/p . Pick a compactum X so that dimQ(X) = 1 and dimZ/p (X) = 2. The

complex L = K(G, 1) is nilpotent and XτK(H1(L), 1) but XτL does not hold. In-

∞ ∞ deed, as π1(L) is notp ¯-local and H1(L; Z/p ) = 0, 5.5.4 says H2(L; Z/p ) 6= 0

which means H2(L)/Tor(H2(L)) is not p-divisible. If XτL, then XτK(H2(L), 2) and

∞ dimZ(p) (X) ≤ 2 and that, in combination with dimQ(X) = 1, implies dimZ/p (X) ≤ 1, a contradiction.

However, if H1(L) is a torsion group, then the answer to 5.1.2 is positive.

Lemma 5.1.3. Suppose N is a nilpotent group. If XτK(Ab(N), 1) for some metrizable space X and Ab(N) is a torsion group, then XτK(N, 1).

Proof. We will prove 5.1.3 by induction on the nilpotency class n of N. Let ΓnN = Γn. Notice N/Γn is a nilpotent group of class (n − 1) whose abelianization is an image of Ab(N). Thus XτK(N/Γn, 1). The epimorphism

⊗nAbN −→ ΓnN = Γn

24 implies XτK(Γn, 1), so the fact that N is a central extension

1 → Γn → N → N/Γn → 1 concludes the proof. For the sake of completeness let us show (2) is always stronger than (3).

Proposition 5.1.4. Suppose X is a metrizable space and L is a connected CW complex. If XτK(πn(L), n) hold for all n ≥ 1, then XτK(Hn(L), n) for all n ≥ 1.

Proof. Let Ln be the CW complex obtained from L by killing all homotopy groups higher than n. Since Ln is obtained from L by attaching k-cells for k > n + 1, Hn(Ln) = Hn(L). Also, one has XτLn as XτK(πi(Ln), i) holds for all i and only ﬁnitely many homotopy groups of Ln are non-trivial (see Theorem G of [25]).

Therefore XτK(Hn(Ln), n).

Deﬁnition 5.1.5. X is called a Knoxville space if it is metrizable and for any connected CW complex L the conditions XτK(πn(L), n) for all n ≥ 1 imply XτL.

Problem 5.1.6. Characterize Knoxville spaces.

It follows from Theorem G of [25] that any ﬁnite dimensional X or any X ∈ ANR is a Knoxville space. Also, it is easy to see that any countable union of closed Knoxville subspaces is a Knoxville space.

5.2 Properties of the homotopy ﬁber of L → SP (L)

Notice that condition (3) of Section 5.1 is equivalent to XτSP (L) as SP (L) is the weak product of K(Hn(L), n) for all n ≥ 1 according to the famous theorem of Dold and Thom [16]. Since we are interested in deriving XτL it makes sense to ponder the stronger condition XτF , where F is the homotopy ﬁber of the inclusion L → SP (L). In this section we concentrate on basic properties of F and its homotopy groups.

25 Proposition 5.2.1. Suppose L is a CW complex and F is the homotopy ﬁber of the inclusion i of L into its inﬁnite symmetric product SP (L). If L is nilpotent, then F is nilpotent.

Proof. The homotopy sequence

j∗ i∗ ∂ · · · → πn(F ) → πn(L) → πn(SP (L)) → πn−1(F ) → · · ·

j i of the ﬁbration F → L → SP (L) is a sequence of π1(L)-modules [43, Proposition

bis 8 .2]. For the action of π1(L) on πn(F ) which is described in the proof of [43,

bis Proposition 8 .2] holds g · α = j∗(g) · α for α ∈ πn(F ) and g ∈ π1(F ).

Let IF and IL be the augmentation ideals of group rings Z[π1(F )] and Z[π1(L)],

c respectively. Because L is a nilpotent, there is an integer c, such that (IL) πn(L) = 0.

c c Let η ∈ (IF ) and α ∈ πn(F ). Then j∗(ηα) = j∗(η)j∗(α) = 0, because j∗(η) ∈ (IL) .

Thus there exists β ∈ πn+1(SP (L)), such that ∂β = ηα. Let g ∈ π1(F ). Then

(j∗(g) − 1) ∈ IL and

∂((j∗(g) − 1)β) = (j∗(g) − 1)∂β = (j∗(g) − 1)ηα(g − 1)ηα.

The action of π1(L) on πn(SP (L)) is deﬁned as lγ = i∗(l)γ for l ∈ π1(L) and γ ∈

πn(SP (L)). Hence

(j∗(g) − 1)β = (i∗j∗(g) − 1)β = (1 − 1)β = 0,

c+1 therefore (g − 1)ηα = 0. This shows that (IF ) πn(F ) = 0, so the space F is

nilpotent.

Proposition 5.2.2. Suppose L is a nilpotent CW complex and F is the homotopy ﬁber of the inclusion i of L into its inﬁnite symmetric product SP (L). If P is a set

of primes such that Hk(L) is a P-torsion group for all k ≤ n, where n ≥ 1 is given,

26 then πk(F ) is a P-torsion group for all 1 ≤ k ≤ n + 1.

Proof. Let P0 be the complement of P in the set of all primes. Consider the

0 localization L(P0) of L at P . Notice that L(P0) is n-connected, so the Hurewicz

homomorphism φk : πk(L(P0)) → Hk(L(P0)) is an isomorphism for k = n + 1 and an

epimorphism for k = n + 2. Let us split the exact sequence ... → πk(F ) → πk(L) →

Hk(L) → ... into ... → π2(F ) → π2(L) → H2(L) → A → 0 and 1 → A → π1(F ) →

B → 1, where B is the commutator subgroup of π1(L). Localizing the ﬁrst sequence

0 at P yields A being a P-torsion group and πk(F ) being P-torsion for 2 ≤ k ≤ n + 1.

Since B is P-torsion, 5.2.2 follows.

Corollary 5.2.3. Suppose L is a nilpotent CW complex and F is the homotopy ﬁber of the inclusion i of L into its inﬁnite symmetric product SP (L). If n > 1 is a number

such that Hk(L) is a torsion group for all k < n, then for any metrizable space X the

conditions XτK(Hk(L), k) for all k ≤ n imply XτK(πk(F ), k) for all 1 ≤ k ≤ n.

Proof. The case k = 1 is taken care of by 5.1.3. If p-torsion of πk(F ) is not

trivial, then 5.2.2 implies that p-torsion of Hm(L) is not trivial for some m < k.

Therefore XτK(Z/p∞, m) and XτK(Z/p, m + 1). This implies XτK(G, k) for all G

in the Bockstein basis of πk(F ) resulting in XτK(πk(F ), k).

5.3 Homotopy groups with coeﬃcients

Given a countable Abelian group G consider a pointed compactum Pn(G) such that its integral cohomology is concentrated in dimension n and equals G. The n-th

homotopy group πn(L; G) of a pointed CW complex L is deﬁned in [45] to be the set

[Pn(G),L] of pointed homotopy classes from Pn(G) to L. If Pn−1(G) exists (that is

always true if n > 2 or G is torsion free and n ≥ 2), then Pn(G) could be taken as

the suspension ΣPn−1(G) of Pn−1(G) with the resulting group structure on πn(L, G).

27 If one puts D = P2(G) (or D = P1(G) if G is torsion-free), then one can analyze

D D homotopy groups of L = Map(D,L) and realize that πn(L ) = πn+2(L; G) (respec-

D tively, πn(L ) = πn+1(L; G)). Therefore, given a Hurewicz ﬁbration F → E → B, one concludes there is a long exact sequence ... → πn(F ; G) → πn(E; G) → πn(B; G) →

D D D πn−1(F ; G) → ... (see [45] for the special case of G = Z/m) because F → E → B is a Serre ﬁbration.

In the special case of G = Z/m one can pick the Moore space D = M(Z/m, 1) for

P2(G). In that case one has a Serre ﬁbration (that follows from the Homotopy Exten- sion Theorem) Map(S2,L) → Map(D,L) → Map(S1,L) where S1 is the 1-skeleton of D and S2 = D/S1. The map Map(D,L) → Map(S1,L) is simply restriction induced.

Since the boundary homomorphism πn+1(B) → πn(F ) in that case amounts to multi-

1 2 plication by m from πn+1(Map(S ,L)) = πn+2(L) to πn(Map(S ,L)) = πn+2(L), one concludes the following (see [45] for another way of deriving an equivalent result):

Lemma 5.3.1. Let D = M(Z/m, 1) for some m ≥ 2. For each pointed CW complex L and each n ≥ 0 one has a natural exact sequence

D 0 → πn+2(L) ⊗ Z/m → πn(L ) → πn+1(L) ∗ Z/m → 0,

m where π1(L) ∗ Z/m is {x ∈ π1(L)|x = 1}.

We are interested in homotopy groups with coeﬃcients in Z/p∞, the direct limit

2 ∞ of Z/p → Z/p → .... Notice that one can construct P2(Z/p ) as the inverse limit of ... → M(Z/pn+1, 1) → M(Z/pn, 1) → ... → M(Z/p, 1) which can be viewed as ˆ ˆ M(Zp, 1), the Moore space for the p-adic integers Zp in terms of Steenrod homology. In that case 5.3.1 becomes

Lemma 5.3.2. Let p be prime. For each pointed CW complex L and each n ≥ 0 one

28 has a natural exact sequence

∞ ∞ ∞ 0 → πn+2(L) ⊗ Z/p → πn+2(L; Z/p ) → πn+1(L) ∗ Z/p → 0,

∞ pk where π1(L) ∗ Z/p is {x ∈ π1(L)|x = 1 for some k ≥ 1}.

As a consequence of 5.3.1, 5.3.2, and Dold-Thom Theorem [16] (πn(SP (L)) =

Hn(L)) one can get that πn(SP (L); G) = Hn(L; G) for all n and G = Z/p or G = Z/p∞.

Proposition 5.3.3. Suppose L is a nilpotent CW complex whose fundamental group is p¯-local for some prime p. Let F be the homotopy ﬁber of the inclusion i : L →

SP (L) of L into its inﬁnite symmetric product. If Hk(L, Z/p) = 0 for k ≤ n, where n ≥ 1, then πk(F, Z/p) = 0 for 2 ≤ k ≤ n + 1 and πk(L, Z/p) = 0 for 2 ≤ k ≤ n.

Proof. Let L˜ be the universal cover of L and let π : L˜ → L be the covering projection. Recall that every nilpotent CW complex L has the p-completion Lp with the map L → Lp inducing isomorphisms of all homology groups with coeﬃcients in

Z/p such that one has a natural exact sequence

∞ ∞ 0 → Ext(Z/p , πn(L)) → πn(Lp) → Hom(Z/p , πn−1(L)) → 0

∞ for all n ≥ 1 (see [[34], Theorem 3.7 on p.416]). By 5.5.2 and 5.5.3 one has Ext(Z/p , π1(L)) = ∞ ˜ˆ ˆ Hom(Z/p , π1(L)) = 0, so the induced map Lp → Lp is a homotopy equivalence. Therefore it induces isomorphism of homology mod p and π induces isomorphism ˜ ˜ of homology mod p. However, H2(L; Z/p) = π2(L; Z/p) = π2(L; Z/p) and π3(L) → ˜ H3(L) is an epimorphism resulting in π3(L; Z/p) → H3(L; Z/p) being an epimorphism. By exactness of mod p groups of a ﬁbration one gets π2(F, Z/p) = 0. That proves 5.3.3 for n = 1. ˜ ˜ If n > 1 we apply mod p Hurewicz Theorem of [45] to L to conclude πn+1(L; Z/p) →

29 ˜ ˜ ˜ Hn+1(L; Z/p) is an isomorphism and πn+2(L; Z/p) → Hn+2(L; Z/p) is an epimorphism. Consequently, πn+1(L; Z/p) → Hn+1(L; Z/p) is an isomorphism and πn+2(L; Z/p) →

Hn+2(L; Z/p) is an epimorphism. Thus πn+1(F ; Z/p) = 0.

Corollary 5.3.4. Suppose L is a nilpotent CW complex whose fundamental group is p¯-local for some prime p. Let F be the homotopy ﬁber of the inclusion i : L → SP (L)

∞ of L into its inﬁnite symmetric product. If Hk(L, Z/p ) = 0 for k ≤ n, where n ≥ 2,

∞ ∞ then πk(L, Z/p ) = πk(F, Z/p ) = 0 for 2 ≤ k ≤ n.

Proof. Case 1: n > 2. Notice Hk(L, Z/p) = 0 for k ≤ n − 1 resulting in

∞ πk(F, Z/p) = 0 for 2 ≤ k ≤ n. Hence πk(F, Z/p ) = 0 for 2 ≤ k ≤ n and from an

∞ exact sequence we get πk(L, Z/p ) = 0 for 2 ≤ k ≤ n. Case 2: n = 2. Let L˜ be the universal cover of L and let π : L˜ → L be the ˜ˆ ˆ covering projection. Notice that the induced map Lp → Lp is a homotopy equivalence. Therefore it induces isomorphism of homology mod p and π induces isomorphism of

˜ ∞ ∞ ˜ homology mod p. However, H2(L; Z/p ) = π2(L; Z/p ) and π3(L) → H3(L) is an

∞ ∞ epimorphism resulting in π3(L; Z/p ) → H3(L; Z/p ) being an epimorphism. By

∞ ∞ exactness of mod p groups of a ﬁbration one gets π2(F, Z/p ) = π2(L, Z/p ) = 0.

Proposition 5.3.5. Suppose L is a nilpotent CW complex, F is the homotopy ﬁber of the inclusion i : L → SP (L) of L into its inﬁnite symmetric product, and n ≥ 2. If

Hk(L, Z(P)) = 0 for k ≤ n, then πk(F, Z(P)) = 0 for 2 ≤ k ≤ n+1 and πk(L, Z(P)) = 0 for 2 ≤ k ≤ n.

0 Proof. Let P be the complement of P in the set of all primes. If Hk(L, Z(P)) = 0

0 0 for k ≤ n, then H1(L) is a P -torsion group resulting in π1(L) being a P -torsion group. Let L(P) be the P-localization of L. It is n-connected, so by the classical

Hurewicz Theorem πk(L(P)) → Hk(L(P)) is an isomorphism for k ≤ n + 1 and an epimorphism for k = n + 2. That corresponds to πk(L; Z(P)) → Hk(L; Z(P)) being an

30 isomorphism for k ≤ n + 1 and an epimorphism for k = n + 2. In view of exactness

of ... → πk(F ; Z(P)) → πk(L; Z(P)) → Hk(L; Z(P)) → ..., 5.3.5 follows.

5.4 Main results

Lemma 5.4.1. Suppose X is a metrizable space, G is an Abelian group, and n ≥ 1.

If dimG(X) > n, then one of the following conditions holds:

a. dimQ(X) ≥ n + 1 and G is not a torsion group.

∞ ∞ b. There is a prime p such that G⊗Z/p 6= 0 and dimQ(X) ≤ n, dimZ/p (X) ≥ n.

∞ c. There is a prime p such that G is p-divisible, Torp(G) 6= 0, and dimZ/p (X) ≥ n + 1.

d. There is a prime p such that Torp(G) is not p-divisible and dimZ/p(X) ≥ n + 1.

Proof. Let dimG(X) = m. Suppose none of (a)-(d) holds. According to Part

(b) of Theorem B of [25] one has m = dimG/Tor(G)(X) or m = dimTor(G)(X). If dimTor(G)(X) ≥ n + 1, then, according to Part (a) of Theorem B of [25], there is a prime p such that either Tor(G) is p-divisible, Torp(G) 6= 0, and dimTor(G)(X) =

∞ dimZ/p (X) (in which case (c) holds) or Tor(G) is not p-divisible, Torp(G) 6= 0, and dimTor(G)(X) = dimZ/p(X) (in which case (d) holds). Therefore m = dimG/Tor(G)(X) and dimTor(G)(X) ≤ n. In particular G is not a torsion group, so dimQ(X) ≤ n as (a) fails to hold.

Consider P = {p|G ⊗ Z/p∞ 6= 0}, the set of primes p such that G/Tor(G) is

not p-divisible. It is shown in [25] (Part (f) of Theorem B) that dimZ(P) (X) ≥

∞ dimG/Tor(G)(X), so dimZ(P) (X) ≥ m. As (b) does not hold, one has dimZ/p (X) ≤ n − 1 for all p ∈ P. From the exact sequence

M ∞ 0 → Z(P) → Q → Z/p → 0 p∈P

31 L ∞ one concludes that the homotopy ﬁber of K(Z(P), m−1) → K(Q, m−1) is K( Z/p , m− p∈P L ∞ 2). Since m − 2 ≥ n − 1, XτK( Z/p , m − 2) which implies XτK(Z(P), m − 1) as p∈P

XτK(Q, m − 1) is true. Thus dimZ(P) (X) ≤ m − 1, a contradiction.

Theorem 5.4.2. Suppose L is a nilpotent CW complex and F is the homotopy ﬁber of the inclusion i of L into its inﬁnite symmetric product SP (L). If X is a metrizable space such that XτK(Hk(L), k) for all k ≥ 1, then XτK(πk(F ), k) and

XτK(πk(L), k) for all k ≥ 2.

Proof. Suppose n ≥ 2 is the smallest natural number such that XτK(πk(F ), k)

fails (similar argument in case XτK(πk(L), k) fails). By 5.4.1 one of the following cases holds for G = πn(F ):

a. dimQ(X) ≥ n + 1 and G is not a torsion group.

∞ ∞ b. There is a prime p such that G⊗Z/p 6= 0 and dimQ(X) ≤ n, dimZ/p (X) ≥ n.

∞ c. There is a prime p such that G is p-divisible, Torp(G) 6= 0, and dimZ/p (X) ≥ n + 1.

d. There is a prime p such that Torp(G) is not p-divisible and dimZ/p(X) ≥ n + 1.

Case 1: dimQ(X) ≤ n−1. Now only (b)-(d) are possible. Let p be the prime from

∞ one of those cases. Notice Hk(L) isp ¯-local for k ≤ n − 1 as otherwise dimZ/p (X) ≤

n − 1 and dimZ/p(X) ≤ n so none of (b)-(d) would be valid. Another observation ∞ ∞ is Hn(L) ⊗ Z/p = 0. Indeed, Hn(L) ⊗ Z/p 6= 0 leads to Hn(L)/Tor(Hn(L)) not

being p-divisible in which case [[25], Part (d) of Theorem B] implies dimˆ (X) ≤ n Zp

as dimHn(L)(X) ≤ n. Therefore dimZ(p) (X) ≤ n (see Part (e) of Theorem B in [25])

∞ and dimZ/p (X) ≤ max(dimQ(X), dimZ(p) (X) − 1) ≤ n − 1, a contradiction. ∞ π1(L) isp ¯-local by 5.5.4 and G⊗Z/p = 0 by 5.3.4. That means (b) is not possible.

If Hn(L) is not p-divisible, then dimZ/p(X) ≤ n and neither (c) nor (d) would be possible. Thus Hk(L; Z/p) = 0 for k ≤ n resulting in G being p-divisible by 5.3.3. That

32 ∞ means only (c) is possible. In addition, Torp(Hn(L)) = 0. Also Hn+1(L) ⊗ Z/p = 0

∞ (otherwise dimZ(p) (X) ≤ n + 1 and dimZ/p (X) ≤ max(dimQ(X), dimZ(p) (X) − 1) ≤ ∞ n). Thus Hk(L; Z/p ) = 0 for k ≤ n + 1. By 5.3.4 Torp(G) = 0, a contradiction.

Case 2: dimQ(X) > n − 1. By 5.2.2 the group G is P-torsion such that

∞ dimZ/p (X) ≤ n − 1 for all p ∈ P which implies dimG(X) ≤ n, a contradiction.

Corollary 5.4.3. Suppose L is a nilpotent CW complex such that πn(L) = πn+1(L) =

0 for some n ≥ 1. If XτSP (L) for some metrizable space X, then XτK(Hn+1(L), n).

Proof. If n = 1, then Hn+1(L) = 0, so assume n ≥ 2. Notice that πn(F ) =

Hn+1(L), where F is the homotopy ﬁber of i : L → SP (L).

Lemma 5.4.4. Suppose L is a nilpotent CW complex and F is the homotopy ﬁber of the inclusion i of L into its inﬁnite symmetric product SP (L). If XτK(H1(L), 1)

for some metrizable space X and H1(L) is ﬁnitely generated, then XτK(π1(F ), 1)

Proof. If H1(L) is ﬁnitely generated and non-torsion, then X is at most 1- dimensional in which case XτL for all connected CW complexes. Therefore assume

H1(L) is a torsion group and (see 5.2.2) there is an exact sequence 1 → A → π1(F ) →

B → 1 such that A and B are P-torsion groups, where P = {p|Torp(H1(L)) 6= 0}.

∞ Notice H1(L) does not contain Z/p for any prime p, so XτK(A, 1) and XτK(B, 1)

which implies XτK(π1(F ), 1).

Theorem 5.4.5. Let X be a metrizable space such that dim(X) < ∞ or X ∈ ANR. Suppose L is a nilpotent CW complex and SP (L) is its inﬁnite symmetric product. If XτSP (L), then XτL in the following cases:

a. H1(L) is ﬁnitely generated.

b. H1(L) is a torsion group.

33 Proof. a. By 5.4.2 and 5.4.4 one concludes XτK(πn(F ), n) for all n ≥ 1. Theorem G of [25] gives XτF which implies XτL.

b. By 5.4.2 and 5.1.3 one concludes XτK(πn(L), n) for all n ≥ 1. Theorem G of

[25] yields XτL.

5.5 Appendix on nilpotent groups

Lemma 5.5.1. Suppose N is a nilpotent group and p is a prime. Ab(N) is p-divisible if and only if N is p-divisible.

Proof. If N is p-divisible clearly so is its abelianization. We will prove the converse by induction on the nilpotency class n of N. Let ΓnN = Γn. Notice N/Γn is a nilpotent group of class (n − 1) whose abelianization is p-divisible. Thus it is p-divisible. The epimorphism

⊗nAbN −→ ΓnN = Γn implies Γn is p-divisible, so the fact that N is a central extension

1 → Γn → N → N/Γn → 1 concludes the proof.

Lemma 5.5.2. Suppose N is a nilpotent group and p is a prime. The following conditions are equivalent:

a. Ext(Z/p∞,N) = 0,

b. Ext(Z/p∞,N) is p-divisible,

c. N is p-divisible.

34 Proof. (a) =⇒ (b) is obvious. For (c) =⇒ (a) let N be p-divisible. Then so is its abelianization and Proposition 3 of [6] implies Ext(Z/p∞,N) = 0. (b) =⇒ (c) If N is not p-divisible neither is its abelianization by 5.5.1. Therefore, by Proposition 3 of [6] Ext(Zp∞ , AbN) is not p-divisible. Then the six-term exact sequence of Hom and Ext [[4], p.170] implies that Ext(Z/p∞,N) is not p-divisible.

Lemma 5.5.3. Suppose G is a nilpotent group and p is a prime. The following conditions are equivalent:

a. Hom(Z/p∞,G) = 0,

b. Hom(Z/p∞,G) is p-divisible,

c. Hom(Z/p∞,G) ⊗ Z/p∞ = 0

d. G does not contain Z/p∞.

Proof. Note that albeit Bousﬁeld and Kan [4] deﬁned Hom as a space, they showed that it is also the set of the respective homomorphisms. (a) =⇒ (b) and (b) =⇒ (c) are obvious.

(c) =⇒ (b). Notice the p-torsion of Hom(Z/p∞,G) is trivial. Indeed, if i : Z/p∞ → G and ip = 1, then for any a ∈ Z/p∞ we ﬁnd b ∈ Z/p∞ satisfying bp = a. Now, i(a) = i(bp) = ip(b) = 1. If an Abelian group A has no p-torsion and A ⊗ Z/p∞ = 0, then A is p-divisible.

(b) =⇒ (d) Suppose i : Z/p∞ → G is a monomorphism. Given a ∈ Z/p∞

k k ﬁnd k ≥ 1 such that ap = 1 and choose φ : Z/p∞ → G so that i = φp . Now i(a) = φpk (a) = (φ(a))pk = φ(apk ) = φ(1) = 1, a contradiction.

(d) =⇒ (a). Given a non-trivial i : Z/p∞ → G its image is a direct sum of copies

∞ of Z/p , a contradiction.

35 ∞ Lemma 5.5.4. Suppose L is a nilpotent CW complex and p is a prime. If H1(L; Z/p ) =

∞ H2(L; Z/p ) = 0, then π1(L) is p¯-local.

∞ ∞ Proof. In view of H2(L; Z/p ) = 0, H1(L) has trivial p-torsion and H1(L; Z/p ) = ˆ 0 implies H1(L) is p-divisible. So is π1(L) (see 5.5.1). Consider the p-completion Lp ˆ ∞ ˆ ∞ ∞ of L. As π1(Lp) = Ext(Z/p , π1(L)) = 0 and H2(Lp; Z/p ) = H2(L; Z/p ) = 0 one ˆ ∞ gets π2(Lp) ⊗ Z/p = 0 by the Hurewicz Theorem. The exact sequence

∞ ˆ ∞ 0 → Ext(Z/p , π2(L)) → π2(Lp) → Hom(Z/p , π1(L)) → 0

∞ ∞ implies Hom(Z/p , π1(L)) ⊗ Z/p = 0. By 5.5.3 π1(L) isp ¯-local.

36 Chapter 6

Bockstein Theorem for nilpotent groups

6.1 Introduction

Problem 6.1.1. Alexandroﬀ’s Problem: Is there a countable family G of abelian groups such that for any compactum X and any abelian group G, the dimension dimG(X) can be expressed in terms of dimH (X), H ∈ G ?

Problem 6.1.1 was solved by Bockstein [2] in 1956, who singled out the following groups (known now as Bockstein groups):

Deﬁnition 6.1.2. The set B of Bockstein groups is

[ ∞ {Q} ∪ {Z/p, Z/p , Z(p)}, p prime

∞ where Z/p is the p-torsion of Q/Z, and Z(p) consists of the rationals whose denom- inator is not divisible by p.

Theorem 6.1.3. If X is compact, then

0This chapter is based on joint work with Cencelj,Dydak and Vavpetiˇc[9]

37 dimG(X) = max{dimH (X) | H ∈ σ(G)}.

The set σ(G) appearing in the theorem above is known as the Bockstein basis of G, and is deﬁned as follows:

Deﬁnition 6.1.4. Given an abelian group G its Bockstein basis σ(G) is the subset of B deﬁned as follows:

1. Q ∈ σ(G) if and only if Q ⊗ G 6= 0,

2. Z/p ∈ σ(G) if and only if (Z/p) ⊗ G 6= 0,

∞ 3. Z(p) ∈ σ(G) if and only if (Z/p ) ⊗ G 6= 0,

4. Z/p∞ ∈ σ(G) if and only if (Z/p∞) ∗ G 6= 0 (here H ∗ G is the torsion product of groups H and G).

Since Pontryagin’s example (see theorem 2.2.3) demonstrated that the logarithmic law dim(X ×Y ) = dim(X)+dim(Y ), known to be true in the case of euclidean spaces, does not hold for compacta, it was natural to seek formulae describing dimG(X × Y ) in terms of the cohomological dimensions of X and Y . This problem was completely solved by Bockstein as follows:

Theorem 6.1.5. Suppose X and Y are compact. Then,

1. dimZ/p(X × Y ) = dimZ/p(X) + dimZ/p(Y ),

2. dimQ(X × Y ) = dimQ(X) + dimQ(Y ),

∞ ∞ ∞ 3. dimZ/p (X × Y ) = max{dimZ/p (X) + dimZ/p (Y ), dimZ/p(X) + dimZ/p(Y ) − 1},

4. dimZ(p) (X × Y ) = dimZ(p) (X) + dimZ(p) (Y ) if

∞ ∞ dimZ(p) (X) = dimZ/p (X) or dimZ(p) (Y ) = dimZ/p (Y ), and

38 ∞ 5. dimZ(p) (X × Y ) = max{dimZ/p (X × Y ) + 1, dimQ(X) + dimQ(Y )} if

∞ ∞ dimZ(p) (X) > dimZ/p (X) and dimZ(p) (Y ) > dimZ/p (Y ).

6.2 Exact sequences

As we have seen earlier, there is no Eilenberg-MacLane space K(G, n), 1 < n < ∞, for non-Abelian groups G, so dimGX ∈ {0, 1, ∞}. Since dimGX = 0 if and only if X is a discrete space, the only interesting question is if XτK(G, 1) holds.

Lemma 6.2.1. If X is metrizable space and dimG(X) = 0 for some group G 6= 1, then dimH (X) = 0 for any group H.

Proof. If dimG(X) = 0, then K(G, 0) ∈ Æ(X). Because K(G, 0) is a discrete space, any discrete space belongs to Æ(X), in particular K(H, 0) ∈ Æ(X) for any group H, so dimH (X) = 0 for every group H.

Lemma 6.2.2. Let X be a metrizable space. If 1 → K → G → I → 1 is an exact sequence of groups and dimK X, dimGX ≤ 1, then max{dimK (X), dimI (X)} = dimG(X).

Proof. In view of 6.2.1 the only interesting case is that of dimK X = dimGX = 1.

Lemma 6.2.3. Let X be a metrizable space, then dimAb(G)(X) ≤ dimGX for any group G.

Proof. In view of 6.2.1 the only interesting case is that of G non-Abelian and dimGX = 1.

39 6.3 Nilpotent groups

Notation: If G is a group, then Ab(G) is its abelianization. If A, B ⊂ G are subgroups, then the commutator subgroup [A, B] is a group generated with all commutators [a, b], a ∈ A and b ∈ B.

The lower central series {Γn(G)} for a group G is deﬁned as follows: Γ1(G) = G, and Γn+1(G) = [Γn(G),G]. If a group G is nilpotent, then there exists an integer c

such that Γc(G) 6= {1} but Γc+1(G) = {1}. The number c (denoted by h(G)) is called the nilpotency class of the nilpotent group G or its Hirsch length. Abelian groups are nilpotent of Hirsch length 1. By [50, Theorem 3.1], for every n exists an epimorphism

n ⊗ Ab(G) → Γn(G)/Γn+1(G).

c In particular, there is an epimorphism ⊗ Ab(G) → Γc(G). A central extension K → G → I of nilpotent group, for which exists an epimorphism ⊗nAb(G) → K for some n, is called a nilpotent central extension. Thus, for every (nonabelian) nilpotent group G, there exists a nilpotent central extension K → G → I such that Hirsch length of I is less than Hirsch length of G.

Lemma 6.3.1. Let 1 → K → G →π I → 1 be a central extension of nilpotent groups.

a. If K and I are p-divisible then G is p-divisible.

b. If the extension is a nilpotent central extension and G is p-divisible, then K and I are p-divisible.

Proof. Suppose K and I are p-divisible. Let g ∈ G. Then π(g) = ip for some i ∈ I. Letg ¯ ∈ G be such that π(¯g) = i. Then π(¯gpg−1) = 1, sog ¯pg−1 = kp for some k ∈ K. Because k ∈ K ⊂ C(G), g = (¯gk−1)p, so G is p-divisible. If G is p-divisible, then any epimorphic image of G is p-divisible. Thus both I and Ab(G) are p-divisible. As there is an epimorphism ⊗nAb(G) → K, K is also

40 p-divisible.

Lemma 6.3.2. Suppose Pi, i = 1, 2, are two classes of nilpotent groups such that for any nilpotent central extension 1 → K → G → I → 1 where h(I) < h(G) the following conditions hold

a. K and I belong to P1 if G ∈ P1,

b. G ∈ P2 if K,I ∈ P2,

are equivalent. If A ∈ P1 =⇒ A ∈ P2 for all Abelian groups A, then P1 ⊂ P2.

Proof. We prove the implication G ∈ P1 =⇒ G ∈ P2 by induction on Hirsch length of G. Suppose 1 → K → G → I → 1 is a nilpotent central extension of groups

and I is of lower Hirsch length than G. Notice K,I ∈ P1. By inductive assumption

K,I ∈ P2. Therefore G ∈ P2.

Corollary 6.3.3. Let A be an Abelian group and 1 ≤ n ≤ 2. Consider the following statements:

1. He∗(G; A) = 0,

2. Hei(G; A) = 0 for all i ≤ n.

If (1) is equivalent to (2) for all Abelian groups G, then the two statements are equivalent for all nilpotent groups G.

r r Proof. Let P1 (respectively, P2 ) be the class of all nilpotent groups G such that

Hei(G; A) = 0 for all i ≤ n (respectively, Hei(G; A) = 0 for all i) and h(G) ≤ r. Our

r r goal is to prove, by induction on r, that P1 = P2 . It is clearly so for r = 1. Assume

m m P1 = P2 for all m < r. Suppose 1 → K → G → I → 1 is a nilpotent central extension of groups such that

r h(I) < h(G) = r. If G ∈ Pj , then H1(K; A) = 0 which implies Hi(G; A) → Hi(I; A)

41 is an epimorphism for i ≤ 2, so Hi(I; A) = 0 for 1 ≤ i ≤ n. By inductive assumption

Hi(I; A) = 0 for all i ≥ 1. If Hei(K; A) is the ﬁrst non-trivial reduced homology group of K, then Leray-Serre spectral sequence implies Hi+1(I; A) 6= 0, a contradiction.

r r−1 Thus both K and I belong to Pj . Conversely, if K,I ∈ Pj , then (by inductive assumption) they have trivial homology with coeﬃcients in A resulting in G having

r r r trivial homology with coeﬃcients in A and G ∈ Pj . Applying 6.3.2 one gets P1 = P2 .

6.4 Bockstein basis

Notation: If G is a group, then Tor(G) is the subgroup generated by torsion elements of G, Torp(G) is the subgroup generated by all elements of G whose order is a power of p, Fp(G) = G/Torp(G), and F (G) = G/T or(G).

The Bockstein groups are: rationals Q, cyclic groups Z/p of p elements, p-adic

∞ m circles Z/p , and p-localizations of integers Z(p) = { n ∈ Q | n is not divisible by p}, where p is a prime number. We gave the classical deﬁnition of the Bockstein basis of an Abelian group G earlier in this section. Here is a deﬁnition for nilpotent groups which is more convenient in this paper as it allows using localization of short exact sequences of nilpotent groups.

Deﬁnition 6.4.1. Let G be a nilpotent group, then the Bockstein basis σ(G) is deﬁned as follows:

1. Q 6∈ σ(G) if and only if G = Tor(G).

2. Z/p∞ 6∈ σ(G) if and only if G isp ¯-local.

3. Z/p 6∈ σ(G) if and only if G is divisible by p.

4. Z(p) 6∈ σ(G) if and only if F (G) isp ¯-local.

42 Remark 6.4.2. Note that according to the above deﬁnition we have

∞ Z(p) ∈ σ(G) ⇒ Z/p ∈ σ(G) ⇒ Z/p ∈ σ(G).

Corollary 6.4.3. For a nilpotent group G the following statements are equivalent:

1. Q 6∈ σ(G),

2. He∗(G; Q) = 0, and

3. H1(G; Q) = 0.

Proof. The equivalence of (2) and (3) follows from 6.3.3. Let P1 be the class of torsion nilpotent groups and let P2 be the class of nilpotent groups such that

H1(G; Q) = 0 (i.e., Ab(G) is torsion). Use 6.3.2 to conclude P1 = P2.

Corollary 6.4.4. For a nilpotent group G, Z/p 6∈ σ(G) if and only if H1(G; Z/p) = 0.

Proof. Let P1 be the class of p-divisible nilpotent groups and let P2 be the class of nilpotent groups such that H1(G; Z/p) = 0. Use 6.3.2 and 6.3.1 to conclude

P1 = P2.

Corollary 6.4.5. For a nilpotent group G the following statements are equivalent:

1. Z/p∞ 6∈ σ(G),

∞ 2. He∗(G; Z/p ) = 0, and

∞ ∞ 3. H1(G; Z/p ) = H2(G; Z/p ) = 0.

Proof. If Z/p∞ ∈/ σ(G), then G isp ¯-local, so all its integral homology groups are

∞ p¯-local and Hi(G; Z/p ) = 0 for all i ≥ 1, which proves the implication (1) =⇒ (2).

∞ Notice (2) ⇐⇒ (3) by 6.3.3. Indeed, if A is Abelian and H1(A; Z/p ) =

∞ H2(A; Z/p ) = 0, then A must bep ¯-local.

43 Let P1 be the class ofp ¯-local nilpotent groups and let P2 be the class of nilpotent

∞ ∞ groups such that H1(G; Z/p ) = H2(G; Z/p ) = 0. Use 6.3.2 to conclude P1 = P2.

Corollary 6.4.6. For a nilpotent group G the following statements are equivalent:

1. Z(p) 6∈ σ(G),

∞ 2. He∗(F (G); Z/p ) = 0, and

∞ ∞ 3. H1(F (G); Z/p ) = H2(F (G); Z/p ) = 0.

∞ It is obvious that Z/p 6∈ σ(G) if and only if G → G(¯p) is an isomorphism. The

following lemma characterizes Z(p) 6∈ σ(G) via localizations.

Lemma 6.4.7. For a nilpotent group G the following statements are equivalent:

1. Z(p) 6∈ σ(G),

2. G → G(¯p) is an epimorphism.

Proof. Observe that (1) ⇐⇒ (2) in case G is an Abelian group. Indeed, in that

∞ case G(¯p) = G ⊗ Z(¯p) and G → G(¯p) is an epimorphism if and only if G ⊗ Z/p = 0. Since G ⊗ Z/p∞ = F (G) ⊗ Z/p∞, G ⊗ Z/p∞ = 0 if and only if F (G) isp ¯-local.

In view of 6.3.2 it remains to show that both classes P1 = {G | Z(p) 6∈ σ(G)} and

P2 = {G | G → G(¯p) is an epimorphism} are EXACT.

Deﬁnition 6.4.8. The torsion-divisible Bockstein basis σTD(G) of G consists of all

∞ Z/p belonging to σ(G). We set σNTD(G) = σ(G)\σTD(G).

Lemma 6.4.9. If G → I is an epimorphism of nilpotent groups, then σNTD(I) ⊂

σNTD(G).

44 Proof. Suppose Q 6∈ σ(G), then G is a torsion group. So I is a torsion group, hence Q 6∈ σ(I). Let Z/p 6∈ σ(G), then G is p-divisible and then also I is p-divisible.

Let Z(p) 6∈ σ(G), then G → G(¯p) is an epimorphism. Becausep ¯-localization is

an exact functor, the map G(¯p) → I(¯p) is an epimorphism and hence I → I(¯p) is an epimorphism.

Lemma 6.4.10. Let 1 → K → G → I → 1 be a central extension of nilpotent groups, then σ(G) ⊂ σ(K) ∪ σ(I).

Proof. Let Q 6∈ σ(K) ∪ σ(I), then K and I are torsion groups. Hence G is torsion, so Q 6∈ σ(G). Let Z/p 6∈ σ(K) ∪ σ(I), then K and I are p-divisible. By Lemma 6.3.1, also G is p-divisible.

∞ Let Z/p 6∈ σ(K) ∪ σ(I), then K → K(¯p) and I → I(¯p) are isomorphisms. Using Five Lemma and the fact thatp ¯-localization is an exact functor, we conclude that also G → G(¯p) is an isomorphism.

Let Z(p) 6∈ σ(K) ∪ σ(I), then K → K(¯p) and I → I(¯p) are epimorphisms. By the three-lemma G → G(¯p) is also an epimorphism.

Lemma 6.4.11. Let 1 → K → G → I → 1 be a nilpotent central extension of groups.

If Z(p) 6∈ σ(Ab(G)) for some prime p, then Z(p) 6∈ σ(K) and Z(p) 6∈ σ(Ab(I)).

Proof. Assume Z(p) 6∈ σ(Ab(G)). That means the map F (Ab(G)) → F (Ab(G))(¯p) is an isomorphism. Because Ab(G) → Ab(I) is an epimorphism and F is a right exact functor, the map F (Ab(G)) → F (Ab(I)) is an epimorphism. Hence the map

F (Ab(I)) → F (Ab(I))(¯p) is an epimorphism. Its kernel is a p-torsion group, so the

kernel is trivial and the map F (Ab(I)) → F (Ab(I))(¯p) is an isomorphism. That

means Z(p) 6∈ σ(Ab(I)).

45 There exists an epimorphism ⊗nAb(G) → K for some integer n. Because F (Ab(⊗nG)) →

n F (Ab(⊗ G))(¯p) is an isomorphism, in the same way as in the previous paragraph we can prove that F (K) → F (K)(¯p) is an isomorphism. Hence Z(p) 6∈ σ(K).

Lemma 6.4.12. Let G be a nilpotent group, then σNTD(G) = σNTD(Ab(G)) and σ(Ab(G)) ⊂ σ(G).

Proof. The inclusion σNTD(Ab(G)) ⊂ σNTD(G) follows from Lemma 6.4.9.

Let us prove σNTD(G) ⊂ σNTD(Ab(G)). Suppose Q 6∈ σNTD(Ab(G)). Because G is a torsion group if and only if Ab(G) is a torsion group, Q 6∈ σNTD(G).

Suppose Z/p 6∈ σNTD(Ab(G)). Because G is p-divisible if and only if Ab(G) is p-divisible, Z/p 6∈ σNTD(G).

Consider the class P of all nilpotent groups such that Z(p) 6∈ σNTD(Ab(G)) implies

Z(p) 6∈ σNTD(G). P clearly contains all Abelian groups. To show P equals the class N of all nilpotent groups it suﬃces to show (see 6.3.2) that for any nilpotent central extension A → G → G0 such that Hirsch length of G0 is less than h(G), A, G0 ∈ P

0 implies G ∈ P. Assume Z(p) 6∈ σNTD(Ab(G)). By 6.4.11 we conclude Z(p) 6∈ σNTD(G )

0 as G ∈ P and Z(p) 6∈ σNTD(A). By 6.4.10 Z(p) 6∈ σNTD(G).

Let us prove now that σ(Ab(G)) ⊂ σ(G). By Lemma 6.4.9, σNTD(Ab(G)) ⊂

∞ σNTD(G). Suppose Z/p 6∈ σ(G). Then G is uniquely p-divisible, so H∗(G; Z) is uniquely p-divisible. In particular Ab(G) = H1(G; Z) is uniquely p-divisible, hence

∞ Z/p 6∈ σ(Ab(G)).

Theorem 6.4.13. If 1 → K → G → I → 1 is a nilpotent central extension, then σ(G) = σ(K) ∪ σ(I).

Proof. By Lemma 6.4.10, σ(G) ⊂ σ(K) ∪ σ(I).

Let us prove that σ(K) ∪ σ(I) ⊂ σ(G). Suppose Q 6∈ σ(G), then G is a torsion group. Therefore K and I are also torsion groups, so Q 6∈ σ(K) ∪ σ(I).

46 Suppose Z/p 6∈ σ(G), then G is p-divisible. By Lemma 6.3.1, K and I are p- divisible, so Z/p 6∈ σ(K) ∪ σ(I).

∞ Suppose Z/p 6∈ σ(G), then G → G(¯p) is an isomorphism. Becausep ¯-localization is an exact functor, the map K → K(¯p) is a monomorphism and the map Ab(G) →

n (Ab(G))(¯p) is an epimorphism. Because there exists an epimorphism ⊗ AbG → K, the map K → K(¯p) is also an epimorphism, hence it is an isomorphism. By Five

∞ Lemma, also the map I → I(¯p) is an isomorphism, so Z/p 6∈ σ(K) ∪ σ(I).

If Z(p) 6∈ σ(G), then Z(p) 6∈ σ(K) and Z(p) 6∈ σ(I) by 6.4.11 and 6.4.12.

6.5 Bockstein spaces

Deﬁnition 6.5.1. A metrizable space X is called a Bockstein space if dimGX = sup{dimH X | H ∈ σ(G)} for all Abelian groups G.

Remark 6.5.2. In the above deﬁnition observe dimGX is an element of N ∪ {0, ∞} and not only in {0, 1, ∞} as in the case of non-Abelian groups G.

Dranishnikov-Repovˇs-Shchepin [24] showed the existence of a separable metric space X of dimension 2 such that dimZ(p) X = 1 for all primes p. Thus, X is not a

Bockstein space as dimZX = 2 > 1 = sup{dimH X | H ∈ σ(Z)}.

Problem 6.5.3. Is every metric ANR a Bockstein space?

Theorem 6.5.4. A metrizable space X is a Bockstein space if and only if dimZl X = dimˆ X for all subsets l ⊂ of the set of prime numbers. Zl P

ˆ Proof. Since σ( l) = σ( l) for all l ⊂ , dim X = dimˆ X holds for any Z Z P Zl Zl Bockstein space X.

Assume dim X = dimˆ X for all subsets l ⊂ of the set of prime numbers. Zl Zl P

Suppose G is a torsion-free Abelian group G. If Z(p) ∈ σ(G), then Theorem B(d) of [25] says dimˆ X ≤ dimGX. Therefore dimGX ≥ sup{dimH X | H ∈ σ(G)}. Z(p)

47 Suppose sup{dimH X | H ∈ σ(G)} = n and consider l = {p | p · G 6= G}. Theorem

B(f) of [25] says dimG(X) ≤ dimZl X. Since σ(G) = σ(Zl), dimGX ≤ dimdimZl X =

sup{dimH X | H ∈ σ(Zl)} = sup{dimH X | H ∈ σ(G)} = n. That proves dimGX =

sup{dimH X | H ∈ σ(G)} for all torsion-free Abelian groups. The same equality holds for all torsion Abelian groups by Theorem B(a) of [25]. In the case of arbitrary Abelian

groups G, as σ(G) = σ(F (G)) ∪ σ(Tor(G)) and dimG = max(dimF (G)X, dimTor(G)X)

(see Theorem B(b) of [25]) one gets dimGX = sup{dimH X | H ∈ σ(G)} as well.

Remark 6.5.5. Notice it is not suﬃcient to assume dim X = dimˆ X for all Z(p) Z(p)

primes p in 6.5.4. Indeed, the space X in [24] has that property as 1 = dimZ(p) X ≥

dimˆ X ≥ 1 for all primes p. Z(p)

Theorem 6.5.6. Let X be a Bockstein space. If G is nilpotent, then dimG(X) ≤ 1

if and only if sup{dimH (X)|H ∈ σ(G)} ≤ 1.

Proof. 6.2.3 Let P1 be the class of all nilpotent groups and let P2 be the class of

nilpotent groups G such that dimG(X) ≤ 1 if and only if sup{dimH (X)|H ∈ σ(G)} ≤

1. Since P2 contains all Abelian groups, in view of 6.3.2 it suﬃces to show that for

any nilpotent central extension K → G → I the conditions K,I ∈ P2 imply G ∈ P2. It is so if G is Abelian, so assume G is not Abelian. Moreover, as σ(Ab(G)) ⊂ σ(G) by 6.4.12 and dimGX ≤ 1 implies dimAb(G)X ≤ 1 (see 6.2.3), each dimG(X) ≤ 1 and sup{dimH (X)|H ∈ σ(G)} ≤ 1 imply dimAb(G)X ≤ 1, so we may as well assume dimAb(G)X ≤ 1. In view of Lemma 6.2.1 and the fact Ab(G) = 1 implies G = 1, the equivalence of conditions dimG(X) ≤ 1 and sup{dimH (X)|H ∈ σ(G)} ≤ 1 may fail only if dimAb(G)X = 1, so assume dimAb(G)X = 1.

∞ Suppose dimH (X) = n > 1 for some H ∈ σ(K). If H 6= Z/p , then H ∈

σNTD(G) = σNTD(Ab(G)) (Theorem 6.4.13 and Lemma 6.4.12), a contradiction. So

H = Z/p∞ for some prime p and Z/p∞ 6∈ σ(Ab(G)). Hence Ab(G) is p-divisible and

48 this is equivalent to G being p-divisible [8, Lemma 5.1]. Because Z/p∞ 6∈ σ(Ab(G)),

∞ ∞ by Lemma 6.4.5 H1(Ab(G); Z/p ) = 0, so also H1(G; Z/p ) = 0. By Lemma 6.4.5,

∞ ∞ H2(G; Z/p ) 6= 0 as Z/p ∈ σ(G). This implies F (H2(G; Z)) is not p-divisible, so Z(p) ∈ σ(H2(G; Z)). If G is not a torsion group, then Ab(G) is not a torsion group, hence by deﬁnition Q ∈ σ(Ab(G)). Therefore dimQ(X) ≤ 1. Using that fact

∞ and Bockstein Inequalities (BI5, BI6 [41]), we get dimZ/p (X) = dimZ(p) (X) − 1.

Because Z(p) ∈ σ(H2(G; Z)), the dimension dimZ(p) (X) ≤ dimH2(G;Z)(X) ≤ 2 as X is

∞ a Bockstein space and then dimZ/p (X) = dimZ(p) (X) − 1 ≤ 1, a contradiction.

Thus G is a torsion group and is a product of q-groups G = Πq∈PGq. Hence

Ab(G) = Πq∈PAb(Gq). Because G is notp ¯-local, Gp 6= 1, but Ab(G) is uniquely p-divisible, so Ab(Gp) = 1. Therefore Gp is a perfect nilpotent group, but such group is trivial, a contradiction.

Thus dimH X ≤ 1 for all H ∈ σ(K) and dimK X = sup{dimH (X)|H ∈ σ(K)} ≤ 1 as K is Abelian and X is a Bockstein space.

Suppose sup{dimH (X)|H ∈ σ(G)} ≤ 1. By Theorem 6.4.13, σ(G) = σ(K)∪σ(I).

Therefore sup{dimH (X)|H ∈ σ(I)} ≤ 1 and dimI X ≤ 1 as I ∈ P2. Consequently, dimGX ≤ 1 by 6.2.2.

Suppose dimGX ≤ 1. By 6.2.2 one gets dimI X ≤ 1 and sup{dimH (X)|H ∈

σ(I)} ≤ 1 as I ∈ P2. By Theorem 6.4.13, σ(G) = σ(K) ∪ σ(I), hence

sup{dimH (X)|H ∈ σ(G)} = sup{dimH (X)|H ∈ σ(K) ∪ σ(I)} ≤ 1.

Corollary 6.5.7. If X is a Bockstein space and XτL for some nilpotent CW complex

L, then dimπn(L)X ≤ n for all n ≥ 1.

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55 Vita

Atish Jyoti Mitra was born in Calcutta, India. He studied mechanical engineering at Jadavpur University, Calcutta, graduating with a bachelor’s degree in 1992 and worked in professional engineering companies from 1992 to 1997. In 1997 he started graduate studies in the Mechanical, Aerospace and Engineering Sciences Department at the University of Tennessee, Knoxville and graduated with a MS in 2000. Since summer 2000 he pursued a PhD in mathematics with concentration in topology and geometry at the University of Tennessee, graduating in August 2006.